A very long, solid cylinder with radius R has positive charge uniformly distributed throughout it, with charge per unit volume r. (a) Derive the expression for the electric field inside the volume at a distance r from the axis of the cylinder in terms of the charge density r. (b) What is the electric field at a point outside the volume in terms of the charge per unit length l in the cylinder

Explanation:

From the given information.

The charge density of the cylinder can be computed as:

where:

= total charge on cylinder

= density of the cylinder

net volume on cylinder

Considering the charge enclosed by the Gaussian surface; we have:

Q = ρV

Now, determining the volume of the cylinder at a distance r from the axis of the cylinder as follows:

Here;

are in the same direction;

If we replace with and q with ; Then:

By rearrangement;

(b)

Using the same formula:

where;

are in the same direction;

If we replace with and with q;

Then:

Replacing for .

From above = the charge per unit length

∴

the answer is the first electromagnet (a). this is because it has more coils around the ferromagnetic material. increasing the number of coils increases the surface area in which the metal interacts with the current in the coil. this enables more atoms in the metal to become aligned hence becoming more magnetized.